A method for evaluating the accuracy of power system state estimation results based on correntropy

Electrical Power and Energy Systems 60 (2014) 45–52

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A method for evaluating the accuracy of power system state estimation results based on correntropy Ye Guo, Wenchuan Wu ⇑, Boming Zhang, Hongbin Sun Department of Electrical Engineering, State Key Laboratory of Power Systems, Tsinghua University, China

a r t i c l e

i n f o

Article history: Received 27 March 2013 Received in revised form 29 January 2014 Accepted 25 February 2014 Available online 21 March 2014 Keywords: Power systems State estimation Residuals Correntropy Uncertainty

a b s t r a c t Power system state estimation (SE) is a crucial basic function in energy management systems (EMSs) which offers the basic load flow models. It is a critical problem to check whether the SE results are credible enough to be used for online decision making or close-loop control. However, there are few published works focus on this topic till now. The accuracy of SE results can be affected by various factors, many of them are very difficult to quantify. Hence a feasible method is to estimate the accuracy of SE results based on residuals. Due to the huge number of measurements in real power systems, it is necessary to establish a reasonable scalar index that represents the residuals of all measurements and directly indicates the accuracy of the SE results. This paper provides a systematic research on the SE accuracy evaluation problem and proposes a new SE accuracy evaluation index based on correntropy. Some conventional SE accuracy evaluation indices are also introduced and compared theoretically and also through extensive numerical tests. Ó 2014 Elsevier Ltd. All rights reserved.

1. introduction There are mainly three essential problems evolved in power system state estimation: measurement accuracy quantification, SE algorithms and result accuracy evaluation. Extensive researches focus on SE algorithms have been published, especially on the suppression of gross errors in topology [1,2], analog measurements [3–6] and network parameters [7–9]. Some effective studies on the measurement accuracy quantification have also been reported [10,11]. However, there are few systematic researches focus on the SE results accuracy evaluation problem. Since the performances of most functions in EMS depend on the SE results, it is a critical problem to check whether the SE results are credible enough to be used for online decision making or close-loop control. The accuracy of the SE results can be affected by various factors including errors in PT/CT, AC sampling errors, errors from information channels, device malfunctioning and parameter errors, SE algorithm, etc. Many of them are very difficult to quantify. Hence it is very difficult to give a deductive combination for all these factors to evaluate the accuracy of SE results, and a residual-based accuracy estimation method should be employed instead. Generally speaking, the smaller the measurement residuals are, the more accurate the SE result is. Since there are huge number of measure-

⇑ Corresponding author. E-mail address: [email protected] (W. Wu). http://dx.doi.org/10.1016/j.ijepes.2014.02.025 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.

ments in a real power system, the various measurement residuals must be represented by a scalar index to quantify the SE accuracy intuitively. There are several conventional SE accuracy evaluation indices. The most common method is using the first or second order norms of residual vector to build evaluation indices. Such indices sum up the absolute values or square values of the residual vector [12,13]. These indices lack clear physical meanings, and maybe infected by gross errors. In metrology, measurement uncertainty is a textbook method to evaluate the accuracy of SE [14,15]. The calculation of measurement uncertainty is a deductive method, which is very difficult for implementation in large-scale practical power systems. Refs. [16,17] suggest using a threshold value of 3% or 5% of the measurement value as a confidence bound for each measurement. In China, the State Grid Corporation of China (SGCC) proposes an official standard for evaluating the accuracy of the SE, which named as acceptance rate (AR). AR counts the rate for the measurement whose residual is less than a artificial set threshold value. In theoretically, AR is loosely connected with measurement uncertainty, but its crucial threshold values are totally determined by manual experience. This paper provides a systemic research on the problem of SE result accuracy evaluation, but the measurement accuracy quantification and SE algorithms are beyond this paper. An important contribution of this paper is that it proposes a new SE accuracy evaluation method, the scalar index of correntropy (COE), and

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makes a comprehensive comparison of the proposed COE with several common scalar indices, including the mean absolute value of weighted residuals (MAR), mean squares of the weighted residuals (MSR), and acceptance rate (AR). The remainder of this paper is organized as follows. Section 2 provides an introduction to the traditional indices. The correntropy index is introduced in Section 3. Section 4 introduces the comparison method used in this paper. Extensive numerical tests used to verify the performances of the various indices, and their results are described in Section 5.

ai ¼

2.1. Mean absolute value of weighted residuals (MAR) The mean absolute value of weighted residuals (MAR) can be expressed as: m 1X jr i =di j m i¼i

AR ¼

m 1X ai  100% m i¼1

ð5Þ

where

2. Common scalar indices

MAR ¼

ment errors falling in the interval [3UISO, +3UISO] is 99.7%. The corresponding coverage factor k is about 3.0 for c = 99%. For a reasonable SE result, most measurement residuals must fall into the interval defined by the extended measurement uncertainty with an appropriate confidence level or coverage factor. Therefore, the acceptance rate (AR) index can be used to evaluate the accuracy of the SE results:

ð1Þ

where m denotes the number of measurements, ri and di are the measurement residual and standard deviation for the ith measurement. A smaller MAR index indicates more accurate SE results.



1 if jri j < ei 0

else

ð6Þ

ei ¼ kU ISOi stands for the extended measurement uncertainty and defines the confidence interval. The AR index in (6) has been adopted by the SGCC as an official standard for evaluating the SE accuracy. However, since the standard measurement uncertainty UISO is difficult to estimate in practical power systems, the value of the extended measurement uncertainty ei is also difficult to quantify, and is always specified according to manual experience. Although the acceptance rate is derived from measurement uncertainty theory, it is very subjective since the threshold value ei is set manually.

2.2. Mean squares of weighted residuals (MSR) 3. Correntropy based scalar index The mean squares of weighted residuals (MSR) can be defined as:

3.1. Brief introduction

m 1X MSR ¼ ðr i =di Þ2 m i¼i

ð2Þ

A smaller MSR index indicates more accurate SE results. 2.3. Measurement uncertainty and acceptance rate (AR) Measurement uncertainty is a textbook method to measure the accuracy of the SE results in metrology. The International Standards Organization (ISO) has defined two types of measurement uncertainties: standard and extended. Standard measurement uncertainty UISO is defined as [15]:

U ISO ¼ K½U 2A þ U 2B 

1=2

ð3Þ

where UA is type A uncertainty, which is measured by statistical methods, and UB is type B uncertainty, which measured more subjectively using non-statistical methods. K is a multiplier used to obtain the confidence of interest. The calculation of measurement uncertainty is deductive which depends on the measurement standard deviations and artificial justice. In real power systems, however, the measurement standard deviations comprise various factors and it is very difficult to estimate their standard deviations. Although some control centers have their manually specified values for measurement standard deviation, di, they are usually inaccurate. The quantification of type B uncertainty for large-scale real power systems is even more difficult. Hence, the application of standard measurement uncertainty becomes impractical. Instead of standard measurement uncertainty, extended measurement uncertainty can be used to evaluate the accuracy of the SE results. Extended measurement uncertainty is defined as:

Pðjri j < kU ISO Þ ¼ c

ð4Þ

where k is the coverage factor and c is the confidence level, which can be 99%. Assume that the measurement errors are Gaussian distributed, the probability of the measurement errors falling in the interval [UISO, +UISO] is 68.3%, and the probability of the measure-

Correntropy is a generalized similarity measure between two random variables in signal processing [18–20]. Assume that x1 and x2 are two random variables with probability density functions f1 and f2, respectively. Their Renyi’s quadric correntropy can be defined as:

Hðx1 ; x2 Þ ¼  log I R f1 ðxÞf2 ðxÞdx

I¼

ð7Þ

where I is the cross information potential and H(x1, x2) is Renyi’s quadric correntropy of x1 and x2. The probability density function can be estimated from the kernel function based on the samples, and the cross information potential can be estimated as: m X ^I ¼ 1 jðx1i ; x2i Þ m i¼1

ð8Þ

where m is the number of samples, x1i and x2i are the ith sample values for x1 and x2, respectively, and j is the kernel function, which should be symmetric non-negative definite. According to the Moore–Aronszajn theorem [21,22], for any symmetric non-negative definite function j, there exists a corresponding reproducing kernel Hilbert space F defined by mapping u: X ? F with j as a kernel function. This yields:

huðx1 Þ; uðx2 Þi ¼ jðx1 ; x2 Þ

ð9Þ

where hi stands for the inner product, which measures the information potential in Hilbert space. Hence, one can conclude that correntropy in fact maps the random variables from their original space to another Hilbert space in which their cross information potential is calculated. An interesting characteristic is that all of the reproducing kernel Hilbert spaces with definite dimensions are omorphism, so we can choose any symmetric non-negative definite kernel function to construct the reproducing kernel Hilbert space F. The entire analysis can be carried out in space F without knowing its exact meaning. The most commonly used kernel function is the Gaussian kernel:

Y. Guo et al. / Electrical Power and Energy Systems 60 (2014) 45–52

1

m X

jðx1 ; x2 Þ ¼ pffiffiffiffiffiffiffi exp  2pr i¼1

ðx1i  x2i Þ2 2r2

! ð10Þ

where r is the kernel size. According to Silverman’s rule [23], when the measurement errors are Gaussian distributed, the optimal choice for the kernel size ropt is:

ropt ¼



1=ðnþ4Þ 4  m1=ðnþ4Þ  d 2n þ 1

ð11Þ

where n is the dimension of state vector. As discussed, the measurement error distribution is not strictly Gaussian distributed; consequently, (11) is not a strict optimal choice for parameter r. However, some researches have shown that correntropy is insensitive to the value of r as long as its value is within the proper range [19]. In addition, Eq. (11) can be viewed as a reference choice for parameter r. 3.2. Power system state estimation accuracy evaluation based on correntropy In power systems, the cross-information potential and correntropy between the measurement value vector z and the estimated value vector h(x) can be estimated by:

^I ¼ pffiffiffiffi1 2prm

m X

  2 i ðxÞÞ exp  ðzi h 2r2

i¼1

^ ðz; hðxÞÞ ¼  log H

" 1 pffiffiffiffi 2prm

m X

#   2 i ðxÞÞ exp  ðzi h 2r2

ð12Þ

i¼1

Neglecting the constant terms in (12) and converting its value to percentage, a new SE accuracy evaluation index based on correntropy (COE for short) can be defined as: 2

m 1X ðzi  hi ðxÞÞ COE ¼ exp  m i¼1 2r 2

!  100%

ð13Þ

In practical power systems, the dimension of the state vector n has a large value, and the kernel size ropt in (11) can be approximated as

ropt  d

ð14Þ

3.3. Characteristics of the index based on correntropy Except for solid theoretical foundation, the index of COE in (13) has several novel characteristics which guarantee its outstanding performance. (1) Continuous and has clear physical meanings

From (15), one can observe that all of the even-order moment will affect the kernel function in (13). Consequently, COE is a nonparametric estimation method. (3) Suppression the influence of gross errors It should be noticed that all the measurements which belong to the input of SE should be considered in SE accuracy evaluation, including the susceptive measurements identified by the SE program but excluding the artificial pre-filtered measurements (a detailed explanation is presented in Section 4.3). Accurate SE results should clearly distinguish normal measurements and gross errors. Consequently, a residual vector with major very small values and a few large residuals indicates accurate SE results; and another residual vector with a lot of medium values indicates inaccurate SE results. The former vector is called a distinguished residual vector, and the latter is called as an infected residual vector for short. The COE index value with single residual is illustrated in Fig. 1. From 1 one can observe that the COE index curve is very flat in the large residual area, which makes a distinguished residual vector to have a better COE index than an infected residual vector.

3.4. Comparison with MAR, MSR and AR It has been shown that the index of COE has several novel characteristics: It is a continuous index with clear meaning; it is a nonparametric method and can suppress the influence of gross errors. In this section, we will compare COE with other indices including MAR, MSR and AR, to demonstrate its effectiveness. MAR and MSR are continuous indices, but they do not have clear physical meanings. In other words, given a value of MAR or MSR index (0.1 or 0.2), one can hardly determine how accurate the SE result is. On contrary, the value of COE (80%, 90%, etc.) directly indicates the accuracy of SE result. The value of AR is also in percentage and has a clear physical meaning, but this index is not continuous. Furthermore, COE has an important advantage compared with AR: AR uses a threshold value e to distinguish acceptable and unacceptable measurements. However, different acceptable measurements have individual residuals, and so do unacceptable measurements, but these different residuals are not reflected in the AR index. In comparison, COE scores continuously each measurement from 0% to 100% according to their residuals. MSR is a parametric method, and it is very sensitive to gross errors. The MSR index of an infected residual vector maybe better than a distinguished residual vector. On the other hand, MAR, AR and COE are less sensitive to gross errors. Furthermore, AR is very sensitive to the accuracy of the measurement standard deviations due to its crisp division of accept-

It can be observed from (13) that the COE index is continuous. In the index of COE in (13), each measurement will give its own COE index value, which in fact means its marking to the given SE result according to the residual. The COE index in (13) is in fact a summary of the markings of all the measurements. Such markings range from 0% to 100%, which clearly demonstrate how reliable the SE result is. (2) Nonparametric estimation method By expanding COE into a Taylor series:

COE ¼

m X 1 1X ð1Þk 2k ðzi  hi ðxÞÞ  100% m i¼1 k¼1 ð2r2 Þk

47

ð15Þ Fig. 1. COE index value with a single residual.

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Y. Guo et al. / Electrical Power and Energy Systems 60 (2014) 45–52

Similarity of two data sequences

MAE data sequences

S E

S E

S E

R E S 1

R E S 2

R E S 10

S E

S E

S E

P R O 1

P R O 2

P R O 10

Data sequences of the evaluated indices

Add random errors

Real state

Measured Values

Fig. 2. Comparison procedure using the same system.

able and unacceptable measurements. COE is much more insensitive to the measurement standard deviations than AR, since they affect only the kernel size. Since measurement standard deviations are difficult to quantify accurately in real power systems, COE also is more suitable for evaluating the SE accurately from this perspective. A summary of the comparison of MAR, MSR, AR and COE is given in Table 1. We use italic characters to denote the advantageous characteristics. Table 1 shows that in theoretical, COE is a better evaluation index than existing MAR, MSR and AR indices.

developed, and expressed as SE PRO1, SE PRO2, . . ., SE PRO10. Using these ten SE programs, ten groups of SE results can be calculated based on the same group of measurements, expressed as SE RES1, SE RES2, . . ., SE RES10. The data sequences of the accuracy evaluation indices for MAR, MSR, AR, and COE can be calculated using these ten SE results. The data sequence of MAE can also be obtained using Eq. (16). The performances of these accuracy evaluation indices can be evaluated using the similarity between their data sequences and the data sequence of MAE. 4.2. Sequences similarity

4. Method for evaluating the performance of different indices 4.1. Procedure The real states of a real power system are unknown, and it is difficult to judge which index provides the most reasonable evaluation result. Therefore, test systems for which the real states can be known should be used to compare the performance of these indices. The mean absolute estimated error (MAE) of the SE results of test systems can be calculated using the differences between the real and estimated values of state variables:

MAE ¼

N 1X jV_ se  V_ real i j N i¼1 i

ð16Þ

_ real are the estiwhere N denotes the bus number, and V_ se i and V i mated and real values for the complex voltage on the ith bus, respectively The comparison procedure used here is illustrated in Fig. 2. The real states for a given test system are known. A group of measurements can be generated by adding random measurement errors (possibly containing some manually set gross errors). Ten SE programs based on the weighted least-square principle with different manual parameters and bad data identification functions were

The similarity of the index and MAE sequences is measured using the length of the longest common sequence (LCS) [24,25]. The LCS is widely used in many disciplines to measure the similarity of two sequences. In this paper, the LCS index is transformed into a percentage using:

LCS index ¼ lLCS =10  100%

ð17Þ

where lLCS is the length of the LCS; the lengths of the index data sequence and data sequence for MAE are both 10 in the comparison procedure. Obviously, the LCS index reflects the similarity between two data sequences. The MAE reflects the true accuracy of the SE results. Therefore, a higher value of the LCS index means that the corresponding evaluation index is more reasonable. 4.3. Measurement set to be considered An important problem in SE accuracy evaluation is which kinds of measurements should be considered. For SE, the erroneous measurements can be identified in two steps including artificial pre-filtering and automatically compressing by bad data identification function as shown in Fig. 3. In the artificial pre-filtering step, some

Table 1 Comparison of MAR, MSR, AR and COE.

Continuity If has clear physical meanings If sensitive to gross errors If sensitive to measurement standard errors

MAR

MSR

AR

COE

Continuous No No No

Continuous No Yes No

Not continuous Yes No Yes

Continuous Yes No No

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Y. Guo et al. / Electrical Power and Energy Systems 60 (2014) 45–52

erroneous measurements are removed according to experiences. The artificial filtered measurements are removed before SE calculation, so they should be excluded in the SE accuracy evaluation. On the other hand, the susceptive measurements identified by the SE program, should be considered in the evaluation, because these measurements in fact have taken part in the SE calculation with bad data identification function. In fact, these identified susceptive measurements may differ from the real gross errors, different SE and bad data identification algorithms have different performances to compress bad data. An extreme example is that one can identify all the measurements as ‘‘susceptive’’ except for a group of key measurements. In such case, all the ‘‘non-susceptive’’ measurements (the rested key measurements) will have zero residuals. If the susceptive measurements are not included in accuracy evaluation, these zero residuals will indicate very accurate SE results erroneously.

Fig. 4. LCS index for AR when e = kd.

5. Numerical tests 5.1. Tests of different values of parameter e for AR In this section, numerical tests are used to find a reasonable choice for parameter e in preparation for comparing the performance of different indices, since this parameter affects the performance of AR significantly. Generally, every measurement might have an individual value for e. For convenience, however, the standard deviations of all of the power measurements are set equal and the standard deviations for voltage amplitude measurements are set to 0.1 times the power measurements in the subsequent tests. Consequently, the same e is used for all of the power measurements, and the e for voltage amplitude measurements is set to 0.1 times the value for the power measurements. The IEEE 118-bus system was used for the tests. The gross error rate was set to 3%, and no parameter or topology errors exist in the tests. Let the standard deviation d of the power measurements change from 0.001 to 0.1; then, the LCS index curves with variable e = kd and constant e are plotted in Figs. 4 and 5, respectively. To realize a comprehensive comparison with statistical meaning, the procedure in Fig. 2 was repeated 100 times, and each point in Figs. 4 and 5 is the average LCS index for these 100 tests. From Figs. 4 and 5, we can conclude that the value of e affects the performance of AR significantly. On average, the LCS index for e = kd is greater when e takes a fixed value, the LCS index is relatively high when e = 4d. Therefore, e = 4d is a reasonable choice for setting the value of e.

Fig. 5. LCS index for AR when e takes a fixed value.

TR#1

5

4

7 2 TR#2

1

6

8 9 TR#3 3 Fig. 6. Configuration and measurement distribution of 9-bus system.

5.2. Illustration on 9-bus system To clearly demonstrate the advantages of the proposed COE index, the 9-bus system in Fig. 6 is studied in this section as an example. Linear SE model is used here, in which only use active power measurements and consider bus voltage angles as state

variables. The standard deviations of measurements are set as 1.50 MW. Single measurement set and two groups of SE results are constructed in this test. The measurements are shown by black square in Fig. 6. For each measurement, its real value, measurement value, error are shown in Table 2. The power injection of

Effective measurements

Effective measurements

Artificial pre-filtering

Full measurement Set

Identified erroneous measurements

SE Calculation with bad data identification function

Artificial filtered measurements Fig. 3. Measurements flow in the whole SE procedure.

Artificial filtered measurements

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Y. Guo et al. / Electrical Power and Energy Systems 60 (2014) 45–52

Table 2 Measurement estimation values, residuals and indices of two groups of SE results for 9-bus system. Meas.

Real value

Meas. value

P1–4 P4–5 P4–6 P2–7 P7–5 P7–8 P3–9 P9–6 P9–8 P5 P6 P8

66.96 38.01 28.95 163.04 86.99 76.05 85.00 61.05 23.95 125.0 90.0 100.0

65.48 39.22 27.15 161.22 87.21 75.12 84.32 58.33 24.77 126.2 70.0 97.6

SE results A

SE results B

SE value

Residual

If acceptable

COE

SE value

Residual

If acceptable

COE

65.67 43.31 22.24 160.39 83.46 76.58 79.45 52.41 26.01 126.7 74.65 102.59

0.19 4.09 4.91 0.83 3.75 1.46 4.87 5.92 1.24 0.50 4.65 4.99

Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

0.9920 0.0242 0.0047 0.8580 0.0439 0.6227 0.0051 0.0004 0.7105 0.9459 0.0081 0.0039

66.22 39.38 26.85 161.80 87.04 74.75 84.11 59.94 24.16 126.42 86.79 98.91

0.74 0.16 0.30 0.58 0.17 0.37 0.21 1.61 0.61 0.22 16.79 1.31

Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes

0.8854 0.9943 0.9801 0.9279 0.9935 0.9700 0.9902 0.5621 0.9206 0.9893 0.0000 0.6829

Table 3 Measurement estimation values, residuals and indices of two groups of SE results for 9-bus system.

SE result A SE result B

Mean error of state variables dh

MAR

MSR

AR (%)

COE (%)

0.0047 7.44e4

1.3022 0.9383

6.1048 10.6608

100 91.67

35.17 82.47

bus 6 is set as a gross errors, and is highlighted by bold characters. For the two groups of SE results, their estimated values, residuals, if determined as acceptable by AR, and the value of COE are also listed in Table 2. The overall indices values for these two groups of SE results are shown in Table 3: The second column in Table 3 gives the mean error of the bus voltage angles, defined by

dh ¼

 n  1X  SE  hi  hReal  i n i¼1

each measurement which clearly shows its accuracy. As a result, the COE index for B is much higher than A, which correctly suggests that B is much more accurate. MAR also suggests that B is more accurate than A, but these values (1.3022 and 0.9383) do not have clear physical meanings and

ð18Þ

where n is the bus number, the superscripts SE and Real denote estimated value and real value, respectively. It can be observed that the error of SE result A is much larger than that of SE result B, indicating B is much more accurate than A. In fact, according to Table 2, it can be concluded that A failed to suppress the gross error and got an infected residual vector (as the 5th column in Table 2). On the other hand, B has successfully suppressed the gross error and got a distinguished residual vector (9th column). However, on the issue of evaluating indices, MSR and AR give erroneous determination that A is more accurate than B. For MSR, it is sensitive to gross errors and the infected residual vector has a better index. For AR, its disadvantage is the equal treatment for all the acceptable measurements. The residual for SE result A has some large value, but none of them violates the threshold value (4d = 6.0 MW). On contrary, COE gives continuous markings for

Fig. 7. Comparison of the different evaluation indices on a 118-bus system.

Fig. 8. Comparison of the different evaluation indices on a 145-bus system.

Fig. 9. Scatter plot for the AR index in the 118-bus system test.

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Y. Guo et al. / Electrical Power and Energy Systems 60 (2014) 45–52 Table 4 Average LCS index declines. AR (%)

COE (%)

MAR (%)

1.55371

0.907143

0.87500

Fig. 10. Scatter plot for the COE index in the 118-bus system test. Fig. 13. The LCS index declines when d is inaccurate.

seemed to be similar. While COE clearly demonstrates that B is quite accurate and A is very inaccurate. In summary, this example clearly shows the conventional indices have their disadvantages which may lead to a erroneous accuracy evaluation. On contrary, the proposed COE does not have such disadvantages, and that is why it can be more reasonable than conventional indices.

3.0

2.5

MAR index

2.0

5.3. Performance comparison among different evaluation indices

1.5

1.0

0.5

0

0.005

0.01

0.015

0.02

MAE (p.u.) Fig. 11. Scatter plot for the MAR index in the 118-bus system test.

300

250

MSR index

200

150

100

General comparisons of the four indices AR, COE, MAR, and MSR were carried out on 118- and 145-bus systems [26]. For the AR index, the threshold value e was set as 4d, based on the previous test results, and the kernel size r for COE was set to equal to d according to (13). Using the procedures in Fig. 2 and 100 repetitions for each point, the LCS curves for these four indices are plotted in Figs. 7 and 8 for the 118- and 145-bus systems, respectively. Figs. 7 and 8 show that COE had the largest LCS index, which indicates that it is the most reasonable index for evaluating the SE accuracy. This is consistent with the theoretical analysis. AR and MAR are suboptimal options, while MSR has the smallest LCS index, making it a poor SE accuracy evaluation standard. Scatter plots are shown in Figs. 9–12 to illustrate the relationship between MAE and the four evaluation indices on the IEEE 118-bus system. The points are concentrated most densely in the scatter plot for COE, which indicates that COE is the most reasonable of the four indices. The shape of the scatter plot for AR is similar to that for COE, but the points are more dispersed. The scatter plot for MAR is similar to that for AR flipped vertically. The samples in MSR were the most widely dispersed, with some outliers, which means MSR cannot indicate the accuracy of the SE results in a robust manner.

5.4. Sensitivity analysis

50

0

0.005

0.01

0.015

MAE (p.u.) Fig. 12. Scatter plot for the MSR index in the 118-bus system test.

0.02

According to (1), (2), (5), and (13), the performance of all four indices depends on the measurement standard deviations d, which are very difficult to quantify accurately in practical power systems. Therefore, it is necessary to check the sensitivity of these indices to d. If the index is very sensitive to d, its performance will deteriorate significantly in a practical application when the standard deviations are inaccurate.

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To verify how much the LCS index is affected by the inaccuracy of d, a normally distributed random error was added to d in this test:

  d0 ¼ 1 þ Nð0; 0:52 Þ  d

ð19Þ

where N(0, 0.52) denotes a normally distributed number with 0 expectation and 0.5 standard deviation. Since in practice, the accurate value of d is unknown, when calculating the index values and their index sequences, only an inaccurate d0 can be used. Under such conditions, the LCS index will fall due to the inaccuracy of d. Repeating the comparison procedure in Fig. 2 based on the 118-bus system, the LCS index declined compared with the results in Fig. 7 for the AR, COE, and MAR indices, which are plotted in Fig. 13 for comparison. Table 4 lists the average declines in the LCS index for these three indices. From Fig. 13 and Table 4, when d had an inaccuracy of about 50%, the LCS index for AR decreased by about 1.55% on average; this was the most sensitive parameter to the accuracy of d. The performances of COE and MAR were much more robust. Considering the LCS index and its sensitivity, COE is the most practical SE accuracy evaluation standard. 6. Conclusions It is important to evaluate the accuracy of the SE results for practical power systems. This is a technical challenge because the real states of power systems are unknown. In this paper, we proposed a practical SE accuracy evaluation index based on correntropy (COE). Theoretical analysis has shown that the proposed index is more reasonable than traditional indices, such as the AR, MAR, and MSR. Extensive numerical tests were used to compare the performances of these evaluation indices, and the results proved that correntropy most reasonably indicates the true accuracy of the SE. AR is another evaluation index, but its performance depends heavily on the accuracy of the measurement standard deviations, which are usually inaccurate for power systems. COE is much less sensitive to the accuracy of the standard deviation measurements than AR, making it more suitable for practical applications. References [1] Lefebvre S, Prévost J. Topology error detection and identification in network analysis. Int J Elec Power Energy Syst 2006;28(5):293–305. [2] Yang T, Sun HB, Bose A. Transition to a two-level linear state estimator—Part I: Architecture. IEEE Trans Power Syst 2011;26(1):46–53.

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